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A reconsideration of nucleation theory
Volume28A, number 1
PHYSICS LETTERS
Table 1 Difference in (LL) between solid and gas.
21 October 1968
Table 2 Shell model parameters and u.v. absorption peak shifts (20°K).
~LL% Theory
~LL% Expt.(n)
~LL% Expt.(E)
Argon
1.17
1.05
0.85
Krypton
2.17
1.29
0.64
Argon
31.7
8.0
0.28
+0.09
+0.43
2.00
Krypton 10.03
30.0
7.6
0.36
+ 0.01
+ 0.14
Xenon
23.1
3.4
0.47
- 0.24
- 0.07
Xenon
3.24
0.70
COo(eV) W2(eV2) V3o(eV2) X1
in e x p e r i m e n t a l v a l u e s f o r the low p r e s s u r e g a s e s . We h a v e a l s o f i t t e d th e e x p e r i m e n t a l data to the s e m i - p h e n o m e n o l o g i c a l D i c k - O v e r h a u s e r s h e l l m o d e l , in the f o r m d e r i v e d by Doniach and Hugglns [2].
3(n 2 - 1) OTP (n2+2~ ~-
~o~3 = 002o+V3(T)_o~2
+ X1 .
11.725 8.43
Shift
(eV)
Theory Expt
t h e f r e e gas a t o m v a l u e , which should b e c o m p a r e d with ~ 1% o b s e r v e d e x p e r i m e n t a l l y . Howe v e r , t h e f r e e a t o m p o l a r i s a b i l i t y as c a l c u l a t e d by t h i s m e t h o d i s u n r e l i a b l e and it i s t h e r e f o r e probably b e t t e r to e x a m i n e the d e r i v a t i v e d a / d a r a t h e r than ~ d i r e c t l y [10] (theory 0.52: expt.
~ 0.20).
(2)
R e s u l t s f o r B 3 = 12 a r e shown in t a b l e 2. As a c he ck of the mOdel, we h a v e c o m p u t e d the shifts of the l o w e s t u.v. a b s o r p t i o n p e a k s in the s o l i d s f r o m t h e i r gas !values and t h e s e a r e a l s o l i s t e d in t a b l e 2 t o g e t h e r with e x p e r i m e n t a l l y o b s e r v e d shifts [9]. A g r e e m e n t i s p o o r and in f a c t s h i f t s m o r e c o n s i s t e n t with the o b s e r v e d v a l u e s a r e o b t a i n e d i,~r p a r a m e t e r s c o r r e s p o n d i n g to B 3 ~ 8. A t h i r d t i g h t - b i n d i n g a p p r o a c h i s due to K e i l [1], who has c a l c u l a t e d the s t a t i c p o l a r i s a b i l i t y a of an a r g o n a t o m in the solid, by e s t i m a t i n g the contribution o f each e l e c t r o n s h e l l to a by m e a n s of a l i n e a r v a r i a t i o n a l technique. T h e t h e o r y p r e d i c t s a d e c r e a s e in ct at T P of about 3.5% f r o m
R~fe~e~ces 1. T.H.Keil, J. Chem. Phys. 46 (1967) 4404. 2. S. Doniach and R. Huggins, Phil. Mag. 12 (1965) 393. 3. R.Mazo, J. Am. Chem. Soc. 86 (1964) 3470. 4. G.O.Jones and B.L.SmRh, Phil. Mag. 5 (1960) 355. 5. A.J.Eatwell and G.O.Jones, Phil, Mag, I0 (1964) 1059. 6. B. L. Smith and C . J . Pings, J. Chem. Phys. 38 (1963) 825. 7. B.L.Smith, Rev. Sci. Inst. 34 (1963) 19. 8. R.L.Amey and R.H.Cole, J. Chem, Phys. 40 (1964) 146. 9. G,Baldini, Phys. Rev. 128 (1962) 1562. 10. B.L.Smith and C.J.Pings, J. Chem. Phys. 48 (1968) 2387.
* * * * *
A RECONSIDERATION
OF
NUCLEATION
THEORY
A. G. BASHKIROV
Institute f o r Scientific Information, Ac. Sci. Moscow, USSR Received 3 September 1968
The reconsideration by Lothe and Pound of the classical nucleation theory is discussed by the method of Brownian motion theory. Their correction factor of 10 17.is found to be the result of having not taken into consideration the nucleus interaction with g a s - c a r r i e r molecules.
One of the m a i n r e s u l t s of the c l a s s i c a l n u c l e ation t h e o r y of G i b b s - V o l m e r - F r e n k e l [1] i s the equation fo r the c o n c e n t r a t i o n of n u c l e i
c* = ¢1 exp { - ~ q * / k T }
(1)
w h e r e AG* i s the f r e e e n e r g y of f o r m a t i o n of the c r i t i c a l n u c l e u s , which i s the su m of f r e e e n e r g y
23
Volume 28A, number 1
PHYSICS LETTERS
of t r a n s i t i o n n u c l e u s ' m o l e c u l e s f r o m gas to l i q uid p h as e and the s u r f a c e f r e e e n e r g y . H o w e v e r in 1962 Lothe and Pound [2] t r i e d to p r o v e that the c l a s s i c a l f o r m u l a (1) was w r o n g b e c a u s e it did not take into account the effect of t r a n s l a t i o n a l and r o t a t i o n a l k i n e ti c e n e r g y being t r a n s f e r r e d to a n u c l e u s . T h i s effect a c c o r d i n g to L o t h e Pound g i v e s an additional f a c t o r of 1017 in eq. (1). T h i s c o r r e c t i o n was d i s c u s s e d by many a u t h o r s [3-6]. H e r e this p r o b l e m will be d i s c u s s e d f r o m the point of view of Brownian motion t h e o r y . Lothe and Pound [3] d e r i d e d the nucleation into t h r e e steps: (i) condensation of ni gas m o l e c u l e s into the liquid phase, (ii) m e c h a n i c a l change of this bulk liquid into n c l u s t e r s of equal s i z e , (iii) t h e r m a l i z a t i o n of the d r o p l e t ' s t r a n s l a t i o n a l and r o t a t i o n a l d e g r e e s of f r e e d o m . Steps (i) and (ii) g i v e the c l a s s i c a l r e s u l t (1) but the step (iii) g i v e s the additional f a c t o r exp {-AGk/kT } w h e r e AGk i s one n u c l e u s ' Gibbs f r e e e n e r g y of a t h e r m a l motion of a s y s t e m of nuclei c o n s i d e r e d as an i d eal gas; T i s a t e m p e r a t u r e . Application of this f o r m u l a e f o r the c a lc u l a t io n of the p o s s i b i l ity of the n u c l e u s t h e r m a l i z a t i o n c o r r e s p o n d s to a p u r e n o n - e q u i l i b r i u m p r o b l e m on t h e r m a l r e laxation between s u b s y s t e m s of nuclei and g a s c a r r i e r being r e p l a c e d by an e q u i l i b r i u m p r o b l e m on a p r o b a b i l i t y that one n u c l e u s of an i d e a l n u clei gas with a t e m p e r a t u r e T o v e r c o m e s the "potential b a r r i e r of a t h e r m a l motion f r e e e n e r gy" c o r r e s p o n d i n g to this t e m p e r a t u r e T. The i n c o r r e c t n e s s of such a p p r o a c h is evident, thus it i s n e c e s s a r y to u s e h e r e n o n e q u i l i b r i u m s t a t i s tical mechanics methods. It i s p o s s i b l e to c o n s i d e r this p r o b l e m as a Brownian motion p r o b l e m , b e c a u s e of the n u c l e u s being l a r g e r , h e a v i e r and l e s s c o n c e n t r a t e d than a gas m o l e c u l e . It is known [7] that Brownian p a r t i c l e s d i s t r i b u t i o n r e l a x e s as _3
W (u,t;Uo):
(27rkT(1- e % [ - 2 ~ t ] ) ) 2 ×
(2)
m I u - u o exp (-Bt) I
x exp l-~k~i ~e~[=~)1
where W is a p o s s i b i l i t y f o r a Brownian p a r t i c l e * * * * *
24
21 October 1968
to have a v e l o c i t y u at a m o m e n t t p r o v i d e d the v e l o c i t y being u o at t = 0, ~ i s the f r i c t i o n coefficient, m i s the gas m o l e c u l e m a s s . Thus the Brownian p a r t i c l e ' s d i s t r i b u t i o n b e c a m e M a x w e l lian with a t e m p e r a t u r e T at a t i m e T ~ 1/28. It is known [8] that
= (prra2/M)~ ~r-mm/21rkT
(3)
f o r a s p h e r e with a r a d i u s a and m a s s M, m o v i n g through a gas with a p r e s s u r e p, p r o v i d e d f r e e flight length l being much m o r e than a. H e r e (p = 1 atm) l = 0.5 × 10 -6 cm , a = 10 -7 cm, thus l >> a. Substitution of M = 100 m = 3 × 10"21g into eq. (3) g i v e s ~" ~ 3 × 10 -6. Thus the Brownian motion t h e o r y p r o v e s the i n e v i t a b i l i t y of a t h e r m a l i z a t i o n of the n u c l e u s t r a n s l a t i o n a l motion and a l l o w s to e s t i m a t e a r e l a x a t i o n t i m e f o r this p r o c e s s . The r o t a t i o n a l d e g r e e s of f r e e d o m may be d i s c u s s e d by the s a m e way. Thus the step {iii) of L o t h e - P o u n d r e c o n s i d e r a t i o n has nothing to do with the p r o c e s s of nuc l e a t i o n and only the two f i r s t s t e p s m u s t be c o n s i d e r e d when c a l c u l a t i n g the c o n c e n t r a t i o n c* of n u cl ei . Th e author would like to e x p r e s s Ms s i n c e r e g r a t i t u d e to D. N. Z u b a r e v f o r his i n t e r e s t in this work.
References 1. J. Frenkel, Kinetic theory of liquids (Oxford, 1946)° 2. J. Lothe and G. Pound, J. Chem. Phys. 36 (1962) 2080. 3. J. Hirth and G. Pound. Condensation and evaporation (N.Y.. 1963). 4. H.Reiss and J.Katz, J. Chem. Phys. 46 (1967) 2486. 5. F.Abraham and G.Pound, J. Chem. Phys. 48 (1968) 732. 6. J . F e d e r et al., Adv. Phys. 15 (1966) 111. 7. S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1. 8. R.Kubo, Statistical mechanics (North-Holland, Amsterdam, 1965).
PHYSICS LETTERS
Table 1 Difference in (LL) between solid and gas.
21 October 1968
Table 2 Shell model parameters and u.v. absorption peak shifts (20°K).
~LL% Theory
~LL% Expt.(n)
~LL% Expt.(E)
Argon
1.17
1.05
0.85
Krypton
2.17
1.29
0.64
Argon
31.7
8.0
0.28
+0.09
+0.43
2.00
Krypton 10.03
30.0
7.6
0.36
+ 0.01
+ 0.14
Xenon
23.1
3.4
0.47
- 0.24
- 0.07
Xenon
3.24
0.70
COo(eV) W2(eV2) V3o(eV2) X1
in e x p e r i m e n t a l v a l u e s f o r the low p r e s s u r e g a s e s . We h a v e a l s o f i t t e d th e e x p e r i m e n t a l data to the s e m i - p h e n o m e n o l o g i c a l D i c k - O v e r h a u s e r s h e l l m o d e l , in the f o r m d e r i v e d by Doniach and Hugglns [2].
3(n 2 - 1) OTP (n2+2~ ~-
~o~3 = 002o+V3(T)_o~2
+ X1 .
11.725 8.43
Shift
(eV)
Theory Expt
t h e f r e e gas a t o m v a l u e , which should b e c o m p a r e d with ~ 1% o b s e r v e d e x p e r i m e n t a l l y . Howe v e r , t h e f r e e a t o m p o l a r i s a b i l i t y as c a l c u l a t e d by t h i s m e t h o d i s u n r e l i a b l e and it i s t h e r e f o r e probably b e t t e r to e x a m i n e the d e r i v a t i v e d a / d a r a t h e r than ~ d i r e c t l y [10] (theory 0.52: expt.
~ 0.20).
(2)
R e s u l t s f o r B 3 = 12 a r e shown in t a b l e 2. As a c he ck of the mOdel, we h a v e c o m p u t e d the shifts of the l o w e s t u.v. a b s o r p t i o n p e a k s in the s o l i d s f r o m t h e i r gas !values and t h e s e a r e a l s o l i s t e d in t a b l e 2 t o g e t h e r with e x p e r i m e n t a l l y o b s e r v e d shifts [9]. A g r e e m e n t i s p o o r and in f a c t s h i f t s m o r e c o n s i s t e n t with the o b s e r v e d v a l u e s a r e o b t a i n e d i,~r p a r a m e t e r s c o r r e s p o n d i n g to B 3 ~ 8. A t h i r d t i g h t - b i n d i n g a p p r o a c h i s due to K e i l [1], who has c a l c u l a t e d the s t a t i c p o l a r i s a b i l i t y a of an a r g o n a t o m in the solid, by e s t i m a t i n g the contribution o f each e l e c t r o n s h e l l to a by m e a n s of a l i n e a r v a r i a t i o n a l technique. T h e t h e o r y p r e d i c t s a d e c r e a s e in ct at T P of about 3.5% f r o m
R~fe~e~ces 1. T.H.Keil, J. Chem. Phys. 46 (1967) 4404. 2. S. Doniach and R. Huggins, Phil. Mag. 12 (1965) 393. 3. R.Mazo, J. Am. Chem. Soc. 86 (1964) 3470. 4. G.O.Jones and B.L.SmRh, Phil. Mag. 5 (1960) 355. 5. A.J.Eatwell and G.O.Jones, Phil, Mag, I0 (1964) 1059. 6. B. L. Smith and C . J . Pings, J. Chem. Phys. 38 (1963) 825. 7. B.L.Smith, Rev. Sci. Inst. 34 (1963) 19. 8. R.L.Amey and R.H.Cole, J. Chem, Phys. 40 (1964) 146. 9. G,Baldini, Phys. Rev. 128 (1962) 1562. 10. B.L.Smith and C.J.Pings, J. Chem. Phys. 48 (1968) 2387.
* * * * *
A RECONSIDERATION
OF
NUCLEATION
THEORY
A. G. BASHKIROV
Institute f o r Scientific Information, Ac. Sci. Moscow, USSR Received 3 September 1968
The reconsideration by Lothe and Pound of the classical nucleation theory is discussed by the method of Brownian motion theory. Their correction factor of 10 17.is found to be the result of having not taken into consideration the nucleus interaction with g a s - c a r r i e r molecules.
One of the m a i n r e s u l t s of the c l a s s i c a l n u c l e ation t h e o r y of G i b b s - V o l m e r - F r e n k e l [1] i s the equation fo r the c o n c e n t r a t i o n of n u c l e i
c* = ¢1 exp { - ~ q * / k T }
(1)
w h e r e AG* i s the f r e e e n e r g y of f o r m a t i o n of the c r i t i c a l n u c l e u s , which i s the su m of f r e e e n e r g y
23
Volume 28A, number 1
PHYSICS LETTERS
of t r a n s i t i o n n u c l e u s ' m o l e c u l e s f r o m gas to l i q uid p h as e and the s u r f a c e f r e e e n e r g y . H o w e v e r in 1962 Lothe and Pound [2] t r i e d to p r o v e that the c l a s s i c a l f o r m u l a (1) was w r o n g b e c a u s e it did not take into account the effect of t r a n s l a t i o n a l and r o t a t i o n a l k i n e ti c e n e r g y being t r a n s f e r r e d to a n u c l e u s . T h i s effect a c c o r d i n g to L o t h e Pound g i v e s an additional f a c t o r of 1017 in eq. (1). T h i s c o r r e c t i o n was d i s c u s s e d by many a u t h o r s [3-6]. H e r e this p r o b l e m will be d i s c u s s e d f r o m the point of view of Brownian motion t h e o r y . Lothe and Pound [3] d e r i d e d the nucleation into t h r e e steps: (i) condensation of ni gas m o l e c u l e s into the liquid phase, (ii) m e c h a n i c a l change of this bulk liquid into n c l u s t e r s of equal s i z e , (iii) t h e r m a l i z a t i o n of the d r o p l e t ' s t r a n s l a t i o n a l and r o t a t i o n a l d e g r e e s of f r e e d o m . Steps (i) and (ii) g i v e the c l a s s i c a l r e s u l t (1) but the step (iii) g i v e s the additional f a c t o r exp {-AGk/kT } w h e r e AGk i s one n u c l e u s ' Gibbs f r e e e n e r g y of a t h e r m a l motion of a s y s t e m of nuclei c o n s i d e r e d as an i d eal gas; T i s a t e m p e r a t u r e . Application of this f o r m u l a e f o r the c a lc u l a t io n of the p o s s i b i l ity of the n u c l e u s t h e r m a l i z a t i o n c o r r e s p o n d s to a p u r e n o n - e q u i l i b r i u m p r o b l e m on t h e r m a l r e laxation between s u b s y s t e m s of nuclei and g a s c a r r i e r being r e p l a c e d by an e q u i l i b r i u m p r o b l e m on a p r o b a b i l i t y that one n u c l e u s of an i d e a l n u clei gas with a t e m p e r a t u r e T o v e r c o m e s the "potential b a r r i e r of a t h e r m a l motion f r e e e n e r gy" c o r r e s p o n d i n g to this t e m p e r a t u r e T. The i n c o r r e c t n e s s of such a p p r o a c h is evident, thus it i s n e c e s s a r y to u s e h e r e n o n e q u i l i b r i u m s t a t i s tical mechanics methods. It i s p o s s i b l e to c o n s i d e r this p r o b l e m as a Brownian motion p r o b l e m , b e c a u s e of the n u c l e u s being l a r g e r , h e a v i e r and l e s s c o n c e n t r a t e d than a gas m o l e c u l e . It is known [7] that Brownian p a r t i c l e s d i s t r i b u t i o n r e l a x e s as _3
W (u,t;Uo):
(27rkT(1- e % [ - 2 ~ t ] ) ) 2 ×
(2)
m I u - u o exp (-Bt) I
x exp l-~k~i ~e~[=~)1
where W is a p o s s i b i l i t y f o r a Brownian p a r t i c l e * * * * *
24
21 October 1968
to have a v e l o c i t y u at a m o m e n t t p r o v i d e d the v e l o c i t y being u o at t = 0, ~ i s the f r i c t i o n coefficient, m i s the gas m o l e c u l e m a s s . Thus the Brownian p a r t i c l e ' s d i s t r i b u t i o n b e c a m e M a x w e l lian with a t e m p e r a t u r e T at a t i m e T ~ 1/28. It is known [8] that
= (prra2/M)~ ~r-mm/21rkT
(3)
f o r a s p h e r e with a r a d i u s a and m a s s M, m o v i n g through a gas with a p r e s s u r e p, p r o v i d e d f r e e flight length l being much m o r e than a. H e r e (p = 1 atm) l = 0.5 × 10 -6 cm , a = 10 -7 cm, thus l >> a. Substitution of M = 100 m = 3 × 10"21g into eq. (3) g i v e s ~" ~ 3 × 10 -6. Thus the Brownian motion t h e o r y p r o v e s the i n e v i t a b i l i t y of a t h e r m a l i z a t i o n of the n u c l e u s t r a n s l a t i o n a l motion and a l l o w s to e s t i m a t e a r e l a x a t i o n t i m e f o r this p r o c e s s . The r o t a t i o n a l d e g r e e s of f r e e d o m may be d i s c u s s e d by the s a m e way. Thus the step {iii) of L o t h e - P o u n d r e c o n s i d e r a t i o n has nothing to do with the p r o c e s s of nuc l e a t i o n and only the two f i r s t s t e p s m u s t be c o n s i d e r e d when c a l c u l a t i n g the c o n c e n t r a t i o n c* of n u cl ei . Th e author would like to e x p r e s s Ms s i n c e r e g r a t i t u d e to D. N. Z u b a r e v f o r his i n t e r e s t in this work.
References 1. J. Frenkel, Kinetic theory of liquids (Oxford, 1946)° 2. J. Lothe and G. Pound, J. Chem. Phys. 36 (1962) 2080. 3. J. Hirth and G. Pound. Condensation and evaporation (N.Y.. 1963). 4. H.Reiss and J.Katz, J. Chem. Phys. 46 (1967) 2486. 5. F.Abraham and G.Pound, J. Chem. Phys. 48 (1968) 732. 6. J . F e d e r et al., Adv. Phys. 15 (1966) 111. 7. S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1. 8. R.Kubo, Statistical mechanics (North-Holland, Amsterdam, 1965).